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https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
> > Consider autonomous ODE $y' = f(y)\quad (1)$ which has an invariant set $M$
defined by the equations
$$h (y) = 0 \qquad (2) $$
i.e., assuming that the initial conditions satisfy $h (y (0)) = 0$, the
solution of the ODE satisties $h ( y ( t )) = 0$ for all later times $t\geq 0$. Defined the Jacobian matrix ($y,h\in \mathbb{R}^n$)
$$H(y)=\frac{\partial h}{\partial y}$$
and assume that it has full row rank for all $t$ (in particular, there are
no more equations in (2) than in (1)).
Next we stabilize the vector field, replacing the autonomous (1) by
$$y' = f (y) - \gamma H^T(HH^T)^{-1}h( y )\quad(3) $$
Show that if there is a constant $\gamma_0$ such that
$$|Hf(y)|_2 \leq \gamma_0 |h(y)|_2$$
for all $y$ in the neighborhood of the invariant set $M$ then $M$
becomes asymptotically stable, i.e. $|h(y(t))|$ decreases in $t$ for
trajectories of (3) starting near $M$ , provided that $\gamma\geq \gamma_0$.

I have no clue to prove the claim. Can anyone help me? Thank you in advance !